World Academy of Science, Engineering and Technology
International Journal of Computer, Electrical, Automation, Control and Information Engineering Vol:7, No:5, 2013
Seat Assignment Problem Optimization
Mohammed Salem Alzahrani
Abstract—In this paper the optimality of the solution of an
International Science Index, Computer and Information Engineering Vol:7, No:5, 2013 waset.org/Publication/11335
existing real word assignment problem known as the seat assignment
problem using Seat Assignment Method (SAM) is discussed. SAM is
the newly driven method from three existing methods, Hungarian
Method, Northwest Corner Method and Least Cost Method in a
special way that produces the easiness & fairness among all methods
that solve the seat assignment problem.
Keywords—Assignment Problem, Hungarian Method, Least Cost
Method, Northwest Corner Method, Seat Assignment Method
(SAM), A Real Word Assignment Problem.
according to the candidates preference in ascending order
where number 1 is the most desirable field of study to the
candidate and the lowest desirable has the rank m. The
academic department has specific criteria to evaluate the
applied students.
After receiving whole applications from all candidates, the
department ranks all candidates according to the agreed
criteria in ascending order where the highest rank candidate is
given number 1 and the last candidate is given number n as
shown in the following preference matrix:
TABLE I
REFERENCE MATRIX
I. INTRODUCTION
F
LIGHT seat assignment, scholarship type assignment and
assigning students to school bus are examples of
assignment problems that have special type of restrictions and
constrains, for example Workstation Assignment Problem [1]
is a specific assignment problem that has a facility constraint.
Some existing methods such as Hungarian Method, Northwest
Corner Method & Least Cost Method[2],[3],[5],[6]will give at
least one solution but will not give guarantee of fairness in
assigning the seat to the candidate. However in some cases
industrial experts proposed a solution to an assignment
problem to create a flexible manufacturing system (FMS) [4]
and some other methods such as QuickMach [7] which solve
linear assignment problems that based on the successive
shortest path algorithm.
In this paper, fairness in assigning the seat to candidate is
the main objective that will draw the optimality of the whole
seat assignment problem using Seat Assignment Method
(SAM).
Here a model description will be introduced which will
contain the model setup & model readings, subsequently a seat
assignment problem will be solved using SAM.
II. MODEL DESCRIPTION
A. Model Setup
To understand SAM easily,one unique example for all
situations is considered that is the model of assigning huge
number of students (more than 1000 students) to number of
major seats or fields of study. At the beginning of each
academic year, every educational organization accepts
application for new students into their academic departments
and majors. The educational organization provides a facility to
the students to choose more than one majors, arranged
Dr. Mohammed Salem Alzahrani, College of Science and Art Shaqra
University, Shaqra – Riyadh, Kingdom of Saudi Arabia (e-mail:
[email protected]).
International Scholarly and Scientific Research & Innovation 7(5) 2013
Field
F1
F2
…
Fj
…
Fm
Required
Seats
sT1
P11
P12
…
P1j
…
P1m
1
sT2
P21
P22
…
P2j
…
P2m
1
sTi
Pi1
Pi2
…
Pij
…
Pim
1
sTn
Pn1
Pn2
…
Pnj
…
Pnm
1
Available
Seats
A1
A2
…
Aj
…
Am
N
Fj : is the jth field of study which any student can apply to.
sTi : is the ith candidate of students.
m : is the total number of major fields of study.
n : is the total number of candidates.
Pij : is the preference rank for ith candidate to jth field of study.
Aj : is the number of seats available in the jth field of study
B. Model Readings
The model readings can be extracted right after setting up
the preference matrix and we are able to find the following
readings:
1. The Most Desirable Field of Study
Since the candidates arrange their preference in ascending
order, then finding the maximum value of (∑
for all
1where
1,2, … , ) is granted to the most desirable
field of study.
2. The Most Undesirable Field of Study
Since the candidates arrange their preference in ascending
order, then finding the maximum value of (∑
for all
where
1,2, … ,
) is granted to the most
undesirable field of study.
628
scholar.waset.org/1999.4/11335
World Academy of Science, Engineering and Technology
International Journal of Computer, Electrical, Automation, Control and Information Engineering Vol:7, No:5, 2013
3. The Most Desirable and Undesirable Field of Study
(Unusual Case)
In some unusual cases the same field of study becomes both
most desirable and undesirable. That case is true for
1,2, … , when the maximum value of ∑
for all
1 and ∑
for all
are occurs at the same
value of j.
International Science Index, Computer and Information Engineering Vol:7, No:5, 2013 waset.org/Publication/11335
4. The Mean of the Preference Data Matrix
Since the built matrix is a preference matrix, then each
candidate have the same number of choices or as we are
calling them field of study which is resulting in:
F1 F2 … Fj … Fm sTi Pi1 Pi2 … Pij … Pim ∑
∑
is always constant for values of
1, 2, … , . That constant can be used to find the total sum of
the preference matrix elements. Then it is true that
since the preference matrix elements is fixed and incremental
by one with no repeating for the same candidate (row wise)
then we can guarantee that m is the total number of elements
in the row and it is also the value of the last element.
which is true always
Therefore, the median is simply
regardless whether the number of elements is even or odd.
6. The Mean and Median of the Preference Data Matrix Are
equal
It is always true that the mean and median of the built
preference data matrix are equal. From the previous model
readings mainly readings 4 and 5,
∑
&
. To prove mean and median
∑
are equal, the proveis initiated from
and we
.
reach
∑
where∑
is known as the mth partial sum of
. By
a triangle series that has m terms and total sum of
th
substituting the m partial sum of the triangle series that has a
sum of
into the main formula for the mean we get
∑
∑
∑
1
1
To find the preference matrix mean, then:
2
2
∑
III. SOLVING THE ASSIGNMENT PROBLEM USING SAM
5. The Median of the Preference Data Matrix
To find the median, the preference matrix has to be
arranged in ascending order which is resulting in:
SAM is a mixture of three methods the Hungarian and
Northwest corner& Least Cost method in a special way that
produces the easiness and fairness of allmethods. Here
considering the preference matrix in table I which we have got
after setting up SAM model previously.
To have the final optimal solution to the assignment
problem we are creating new matrix called Result matrix
where Rij has the values 1 or 0. If Rij is 1 then ith candidate is
assigned to jth seat.
sT1 1 2 … J … m sT2 1 2 … J … m sTi 1 2 … J … m Field F 1 F2 … Fj … F m Required Seats sTn 1 2 … J … m sT1 R11 R12 … R1j … R1m 1 sT2 R21 R22 … R2j … R2m 1 sTi Ri1 Ri2 sTn Rn1 Rn2 … Rnj … Rnm 1 Available Seats A1 A2 … Aj … Am n TABLE II
RESULT MATRIX
In none matrix form the data is shown as:
1, 1, … , 2, 2, … , … , … , … , , , … , … … … ,
,
,…
It holds true that each element of the data is recurring n
times. In other words, frequency of each element is exactly n.
So, it is valid that if we find the median for any candidate,
then it holds true that the median for one candidate is equal to
the median for the whole data set. Consequently, the median is
the middle element of the following set: 1, 2, … , , … , and
International Scholarly and Scientific Research & Innovation 7(5) 2013
… Rij … Rim 1 SAM starts from the upper left corner (Northwest corner) of
629
scholar.waset.org/1999.4/11335
International Science Index, Computer and Information Engineering Vol:7, No:5, 2013 waset.org/Publication/11335
World Academy of Science, Engineering and Technology
International Journal of Computer, Electrical, Automation, Control and Information Engineering Vol:7, No:5, 2013
the preference matrix and do the following:
1. Determine capacity for the jth field
for all
1
2. Initialize
0 for all i and j.
3. Set
1,
1 and go to 9
4. Update
1
5. If
, go to 13
6. Set
1 and go to 9
7. Update
1
8. If
, go to 4
9. Is Pij the minimum value row wise in the preference
matrix? If yes go to 10, if no go to 7.
10. Is
0, If yes go to 11, if no go to 7 (seat availability
constraint)
11.
1
12. Update
1 and go to 4
13. End
SAM can use the least cost method with some
modifications and with help of newly created formula. The
least cost method uses the cost value to give priority to each
candidate subjected to the chosen seat availability. Using
SAM will give more intelligence of fairness using modified
formula for cost that brings three main parts
1. Seat field priority Pij that was chosen by the candidate.
2. Number of fieldsm.
3. Seat field availability Aj.
The following Table IIIis the setup for SAM after arranging
seat candidates according to agreed criteria after laying SAM
in format of Least Cost Method in the following tableau:
least cost method, the SAM setup is very handy to use with the
help of newly driver formula
.
To show the advantages & performance of SAM, infuture
paper an application will be introduced which will help an
organization to ensure a very fair solution of seat assignment
problem.
IV.CASE STUDY
This case study is used to illustrate by numbers and figures
the effectiveness of SAM. The case is done on small size set
of date with 20 candidates and 5 types of seat. College of
Science in University x has to admit students on orientation
year first and then give the student chance to choose one of the
5 offered majors (fields of study).
After collecting students' preferences, the college
administrator arrange the students' requests in ascending order
according to their GPA. By that way he builds the preference
matrix as it is shown in table IV.
TABLE IV
REFERENCE MATRIX FOR STUDENTS IN COLLEGE OF SCIENCE
Required
F5
Field
F1
F2
F3
F4
Seats
sT1
3
2
4
1
5
1
sT2
5
3
1
2
4
1
sT3
4
1
5
3
2
1
sT4
5
1
2
3
4
1
sT5
2
4
3
5
1
1
sT6
3
1
2
5
4
1
sT7
1
5
2
4
3
1
sT8
4
1
2
5
3
1
sT9
5
2
3
1
4
1
sT10
3
5
4
2
1
1
sT11
2
3
1
5
4
1
sT12
5
1
4
2
3
1
sT13
1
5
4
3
2
1
sT14
3
2
5
4
1
1
1 sT15
3
5
1
4
2
1
n sT16
3
4
5
2
1
1
sT17
4
1
5
2
3
1
sT18
3
5
1
4
2
1
sT19
5
2
4
3
1
1
sT20
1
3
4
2
5
1
Available
Seats
6
2
4
3
5
20
TABLE III
FORMATTED SAM'S TABLEAU
Field sT1 sT2 F1 C11 P12 C12 1 1 1 2 P21 C21 P22 C22 2
1
2 2 Available Seats … … Fj C1j 1 j P2j C2j 2 j Ci1 Pi2 Ci2 i
1
I 2 i Cn1 Pn2 Cn2 n 1 N 2 A 2 … … … C1m
1 m P2m C2m
2 m … Pim Cim i m Cnj n
j 1 1 Pnj 1 j Aj P1m Pij Cij Pn1 A1 … … Required Seats Fm … P1j Pi1 sTn … P11 sTi F2 … … Pnm Cnm
n m
Am where the new closed formula forthe corner cost is:
.
The resulting Cijgives the 100% of assurance of fairness in
assigning the seats to the candidates.
Hence achieving the fairness in seat assignment problem by
International Scholarly and Scientific Research & Innovation 7(5) 2013
630
scholar.waset.org/1999.4/11335
World Academy of Science, Engineering and Technology
International Journal of Computer, Electrical, Automation, Control and Information Engineering Vol:7, No:5, 2013
After setting up the model, SAM tableau is built as shown:
TABLE VII
FORMATTED SAM'S TABLEAU FOR COLLEGE OF SCIENCE STUDENTS
F1
sT1
sT2
sT3
sT4
sT5
sT6
International Science Index, Computer and Information Engineering Vol:7, No:5, 2013 waset.org/Publication/11335
sT7
sT8
sT9
sT10
sT11
sT12
sT13
sT14
sT15
sT16
sT17
sT18
sT19
sT20
Available Seats 3
1
5
2
4
3
5
4
2
5
3
6
1
7
4
8
5
9
3
10
2
11
5
12
1
13
3
14
3
15
3
16
4
17
3
18
5
19
1
20
F2
3
1
10
1
14
1
20
1
22
1
28
1
31
1
39
1
45
1
48
1
52
1
60
1
61
1
68
1
73
1
78
1
84
1
88
1
95
1
96
1
6 2
1
3
2
1
3
1
4
4
5
1
6
5
7
1
8
2
9
5
10
3
11
1
12
5
13
2
14
5
15
4
16
1
17
5
18
2
19
3
20
F3
2
2
8
2
11
2
16
2
24
2
26
2
35
2
36
2
42
2
50
2
53
2
56
2
65
2
67
2
75
2
79
2
81
2
90
2
92
2
98
2
2 4
1
1
2
5
3
2
4
3
5
2
6
2
7
2
8
3
9
4
10
1
11
4
12
4
13
5
14
1
15
5
16
5
17
1
18
4
19
4
20
F4
4
3
6
3
15
3
17
3
23
3
27
3
32
3
37
3
43
3
49
3
51
3
59
3
64
3
70
3
71
3
80
3
85
3
86
3
94
3
99
3
4 1
1
2
2
3
3
3
4
5
5
5
6
4
7
5
8
1
9
2
10
5
11
2
12
3
13
4
14
4
15
2
16
2
17
4
18
3
19
2
20
1
4
7
4
13
4
18
4
25
4
30
4
34
4
40
4
41
4
47
4
55
4
57
4
63
4
69
4
74
4
77
4
82
4
89
4
93
4
97
4
3 Required
Seats
F5
5
1
4
2
2
3
4
4
1
5
4
6
3
7
3
8
4
9
1
10
4
11
3
12
2
13
1
14
2
15
1
16
3
17
2
18
1
19
5
20
5
5
9
5
12
5
19
5
21
5
29
5
33
5
38
5
44
5
46
5
54
5
58
5
62
5
66
5
72
5
76
5
83
5
87
5
91
5
100
5
5 1
1
1
V. CONCLUSION
Fairness in seat assignment is proposed based on the Seat
Assignment Method (SAM), which gives the optimal solution
with fairness. To setup the SAM model in the least cost
method gives the intelligence of fairness for finding the
solution of seat assignment problems, Therefore SAM can be
used in real word seat assignment problem for optimal
solution.This approach can also be extended efficiently and
effectively to get much fairness in Seat Assignment Problem.
1
1
1
REFERENCES
[1]
[2]
1
1
[3]
[4]
1
[5]
1
[6]
1
1
[7]
1
A Fast, Near Optimal Heuristic for the Work Station Assignment
Problem, Candance Aral Yano.
Applied Management Science: Modeling, Spreadsheet Analysis, and
Communication for Decision Making, 2nd Edition, John A.
Lawrence,Barry A. Pasternack.
Operations Research: An Introduction (9th Edition), Hamdy A. Taha
Operations Research: Methods, Models, and Applications, Jay E
Aronson, Stanley Zionts.
Operations Research Applications and Algorithms 4th Edition, Wayne
Winston, 2003.
Managerial Decision Modeling with Spreadsheets (2nd Edition),
NagrajBalakrishnan (Author), Barry Render (Author), Ralph M. Stair
(Author).
Orlin, James B., 1953- &Lee, Yusin., 1993. "QuickMatch - a very fast
algorithm for the assignment problem,“ Working papers 3547-93.,
Massachusetts Institute of Technology (MIT), Sloan School of
Management.
1
1
1
1
1
1
1
20 By using SAM as new added layer to the Least Cost
Method, it is easy and fair to use it as optimization technique.
TableVIIshows the optimal and fair solution to the problem
using SAM for Least Cost Method after applying the original
Least Cost Method to the modified table VI which has the
SAM's formulais buried in the cost cell for the least cost
tableau. The direct assignment result is shown in Table VII
TABLE VII
SAM'S SOLUTION WITH LEST COST METHOD
Student #
Field
Student #
Field
F4
sT1 F3 sT11 F3
sT2 F4 sT12 F2
sT3 F5 sT13 F2
sT4 F5 sT14 F5
sT5 F5 sT15 F3
sT6 sT16 F1 F1
sT7 F1 sT17 F3
sT8 F1 sT18 F4
sT9 F1 sT19 F5
sT10 F1 sT20 International Scholarly and Scientific Research & Innovation 7(5) 2013
631
scholar.waset.org/1999.4/11335